I am trying to understand how to calculate the Gauss curvature using the formula $K(p)=det(d_pN)$ where $N:M^2\to\mathbb{R}^3$ is a Gauss map.
To wrap my head around it, I tried to use the torus as an example. Using the parametrization $\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}(r\cos{\theta}+R)\cos{\varphi}\\(r\cos{\theta}+R)\sin{\varphi}\\R\sin{\theta}\end{pmatrix}$ where $R>r>0$ are constant, I calculated the cross product of the two columns of the Jacobian, $t=\begin{pmatrix}-R(r\cos{\theta}+R)\cos{\theta}\cos{\varphi}\\-R(r\cos{\theta}+R)\cos{\theta}\sin{\varphi}\\-r(r\cos{\theta}+R)\sin{\theta}\end{pmatrix}$, then $\Vert t\Vert=(r\cos{\theta}+R)\sqrt{R^2\cos^2{\theta}+r^2\sin^2{\theta}}$ to finally get the Gauss map $N=\dfrac{t}{\Vert t\Vert}=\dfrac{-1}{\sqrt{R^2\cos^2{\theta}+r^2\sin^2{\theta}}}\begin{pmatrix}R\cos{\theta}\cos{\varphi}\\R\cos{\theta}\sin{\varphi}\\r\sin{\theta}\end{pmatrix}$.
However now I run into a problem, $N$ is a 3d vector with 2 variables meaning that $d_pN$ is a $3\times2$ matrix so it has no determinant. Could somebody tell me where I went wrong?